<s xml:id="echoid-s2572" xml:space="preserve">K T, y. </s>
<s xml:id="echoid-s2573" xml:space="preserve">Nam quia data eſt curva A B F,
<lb/>
eique B M ad angulos rectos ducta, invenietur inde quanti-
<lb/>
tas lineæ K M, per methodum tangentium à Carteſio traditam,
<lb/>
quæ ipſi K T, ſive y æquabitur, & </s>
<s xml:id="echoid-s2574" xml:space="preserve">ex ea æquatione, natura
<lb/>
curvæ T V innoteſcet, ad quam deinde tangens ducenda
<lb/>
eſt. </s>
<s xml:id="echoid-s2575" xml:space="preserve">Sed clariora omnia fient ſequenti exemplo.</s>
<s xml:id="echoid-s2576" xml:space="preserve"/>
</p>
<p>
<s xml:id="echoid-s2577" xml:space="preserve">Sit A B F paraboloides illa, cui ſuperius rectam æqua-
<lb/>
<note position="left" xlink:label="note-0164-02" xlink:href="note-0164-02a" xml:space="preserve">TAB. XVI.
<lb/>
Fig. 3.</note>
lem invenimus; </s>
<s xml:id="echoid-s2578" xml:space="preserve">in qua nempe cubi perpendicularium in
<lb/>
rectam S K, ſint inter ſe ſicut quadrata ex ipſa S K abſciſ-
<lb/>
ſarum. </s>
<s xml:id="echoid-s2579" xml:space="preserve">Et oporteat invenire curvam C D E cujus evolu-
<lb/>
tione paraboloides S B F deſcribatur.</s>
<s xml:id="echoid-s2580" xml:space="preserve"/>
</p>
<p>
<s xml:id="echoid-s2581" xml:space="preserve">Hic primum ratio B O ad B P facile invenitur, quia
<lb/>
tangentem paraboloidis in puncto B duci ſcimus, ſumpta S H
<lb/>
æquali {1/2} S K. </s>
<s xml:id="echoid-s2582" xml:space="preserve">Cui tangenti cum B M ad angulos rectos in-
<lb/>
ſiſtat, dantur jam lineæ M H, H K, ac proinde earum in-
<lb/>
ter ſe ratio, quæ eſt eadem quæ O B ad B P.</s>
<s xml:id="echoid-s2583" xml:space="preserve"/>
</p>
<p>
<s xml:id="echoid-s2584" xml:space="preserve">Ut autem ratio B P, ſive K L ad M N innoteſcat, po-
<lb/>
nantur ad K L perpendiculares rectæ K T, L V, æquales
<lb/>
ſingulis K M, L N, ſitque V X parallela L K. </s>
<s xml:id="echoid-s2585" xml:space="preserve">Jam quia
<lb/>
ex duabus ſimul K L, L N, auferendo K M, relinquitur
<lb/>
M N ; </s>
<s xml:id="echoid-s2586" xml:space="preserve">hoc eſt, auferendo ex duabus X V, V L, ſive</s>
</p>
<p>
<s xml:id="echoid-s2587" xml:space="preserve">
<lb/>
<note symbol="*" position="foot" xlink:label="note-0164-03" xlink:href="note-0164-03a" xml:space="preserve">In Exemplari ſuo ad marginem ſcripſit Auctor. ſupponitur hic rectam L N
<lb/>
majorem eſſe quam K M, quod melius fuerat antea probari, etſi verum eſt.
<lb/>
Demonſtratio autem haud difficilis eſt, ſit abſciſſa S K = x; perpendicularis K B
<lb/>
= u; Tatus rectum paraboloidis = a. Quia S H = {1/2} SK, eſt H K = {3/2} S K
<lb/>
({3/2}x). Propter angulum rectum H B M, triangula rectangula H B K, K B M
<lb/>
ſimilia ſunt, & H K ({3/2}x), K B (u), K M, ſunt in continua proportione; ergo
<lb/>
K M = {2uu/3x}, cujus quadratum eſt {4u
<emph style="super">4.</emph>
/9xx} = {4au
<emph style="super">4.</emph>
/9axx}; ſed ut notavit auctor ex natu-
<lb/>
ra Paraboloidis A B F, u
<emph style="super">3</emph>
= axx; ergo quadratum lineæ K M = {4au
<emph style="super">4</emph>
/9axx} = {4au
<emph style="super">4</emph>
/9u
<emph style="super">3</emph>
} =
<lb/>
{4/9} a u unde ſequitur ipſam K M, augeri ſi creſcat B K (u). Cum autem L F exce-
<lb/>
dat B K, L N ſuperabit K M, quod demonſtrandum erat.</note>